Spaniard Alonso, 42, has been heavily linked with the role at Anfield since Klopp said he will stand down.
Leverkusen are on the brink of a first Bundesliga title and are unbeaten in all competitions this season.
Bayern Munich’s honorary president Uli Hoeness said a move for Alonso this summer is “probably impossible”.
Alonso was also seen as a candidate to take over at the Bavarian giants, whose manager Thomas Tuchel will depart at the end of the campaign.
“[Alonso] is more inclined to stay at Bayer Leverkusen in view of their current successes, because he would not want to leave them behind,” Hoeness said.
Alonso, a stylish midfielder, played 210 times for Liverpool from 2004 to 2009.
He left to join Real Madrid – another club he has recently been linked with – and later moved to Bayern where he spent three years before retiring from playing.
Alonso moved into coaching with Real Madrid’s under-14s in 2018 and then had three years in charge of Real Sociedad’s B team.
Read Also: Cristiano Ronaldo told he is no longer Portugal’s ‘best player’ as new star takes the title
In October 2022, Leverkusen gave him his first role in first-team football and, having taken over with the club second-bottom, he led them to a sixth-placed finish in the Bundesliga.
This season, Alonso’s first full term in charge, Leverkusen are 10 points clear of champions Bayern with eight games remaining.
They are also in the semi-finals of the German Cup and the quarter-finals of the Europa League.
“Let’s say if he had two or three more years of success, it would probably be easier to bring him out of there,” Hoeness said.
